#include "eigenvaluedecomposition.h"

#include "matrix.h"

#include <cmath>

using namespace std;

using namespace CPAMA;

void EigenvalueDecomposition::tred2 () {

//  This is derived from the Algol procedures tred2 by
//  Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
//  Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
//  Fortran subroutine in EISPACK.

   for (int j = 0; j < n; j++) {
      d[j] = V[n-1][j];
   }

   // Householder reduction to tridiagonal form.

   for (int i = n-1; i > 0; i--) {

      // Scale to avoid under/overflow.

      double scale = 0.0;
      double h = 0.0;
      for (int k = 0; k < i; k++) {
         scale = scale + abs(d[k]);
      }
      if (scale == 0.0) {
         e[i] = d[i-1];
         for (int j = 0; j < i; j++) {
            d[j] = V[i-1][j];
            V[i][j] = 0.0;
            V[j][i] = 0.0;
         }
      } else {

         // Generate Householder vector.

         for (int k = 0; k < i; k++) {
            d[k] /= scale;
            h += d[k] * d[k];
         }
         double f = d[i-1];
         double g = sqrt(h);
         if (f > 0) {
            g = -g;
         }
         e[i] = scale * g;
         h = h - f * g;
         d[i-1] = f - g;
         for (int j = 0; j < i; j++) {
            e[j] = 0.0;
         }

         // Apply similarity transformation to remaining columns.

         for (int j = 0; j < i; j++) {
            f = d[j];
            V[j][i] = f;
            g = e[j] + V[j][j] * f;
            for (int k = j+1; k <= i-1; k++) {
               g += V[k][j] * d[k];
               e[k] += V[k][j] * f;
            }
            e[j] = g;
         }
         f = 0.0;
         for (int j = 0; j < i; j++) {
            e[j] /= h;
            f += e[j] * d[j];
         }
         double hh = f / (h + h);
         for (int j = 0; j < i; j++) {
            e[j] -= hh * d[j];
         }
         for (int j = 0; j < i; j++) {
            f = d[j];
            g = e[j];
            for (int k = j; k <= i-1; k++) {
               V[k][j] -= (f * e[k] + g * d[k]);
            }
            d[j] = V[i-1][j];
            V[i][j] = 0.0;
         }
      }
      d[i] = h;
   }

   // Accumulate transformations.

   for (int i = 0; i < n-1; i++) {
      V[n-1][i] = V[i][i];
      V[i][i] = 1.0;
      double h = d[i+1];
      if (h != 0.0) {
         for (int k = 0; k <= i; k++) {
            d[k] = V[k][i+1] / h;
         }
         for (int j = 0; j <= i; j++) {
            double g = 0.0;
            for (int k = 0; k <= i; k++) {
               g += V[k][i+1] * V[k][j];
            }
            for (int k = 0; k <= i; k++) {
               V[k][j] -= g * d[k];
            }
         }
      }
      for (int k = 0; k <= i; k++) {
         V[k][i+1] = 0.0;
      }
   }
   for (int j = 0; j < n; j++) {
      d[j] = V[n-1][j];
      V[n-1][j] = 0.0;
   }
   V[n-1][n-1] = 1.0;
   e[0] = 0.0;
}


void EigenvalueDecomposition::tql2 () {

//  This is derived from the Algol procedures tql2, by
//  Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
//  Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
//  Fortran subroutine in EISPACK.

   for (int i = 1; i < n; i++) {
      e[i-1] = e[i];
   }
   e[n-1] = 0.0;

   double f = 0.0;
   double tst1 = 0.0;
   double eps = pow(2.0,-52.0);
   for (int l = 0; l < n; l++) {

      // Find small subdiagonal element

      tst1 = fmax(tst1,abs(d[l]) + abs(e[l]));
      int m = l;
      while (m < n) {
         if (abs(e[m]) <= eps*tst1) {
            break;
         }
         m++;
      }

      // If m == l, d[l] is an eigenvalue,
      // otherwise, iterate.

      if (m > l) {
         int iter = 0;
         do {
            iter = iter + 1;  // (Could check iteration count here.)

            // Compute implicit shift

            double g = d[l];
            double p = (d[l+1] - g) / (2.0 * e[l]);
            double r = hypot(p,1.0);
            if (p < 0) {
               r = -r;
            }
            d[l] = e[l] / (p + r);
            d[l+1] = e[l] * (p + r);
            double dl1 = d[l+1];
            double h = g - d[l];
            for (int i = l+2; i < n; i++) {
               d[i] -= h;
            }
            f = f + h;

            // Implicit QL transformation.

            p = d[m];
            double c = 1.0;
            double c2 = c;
            double c3 = c;
            double el1 = e[l+1];
            double s = 0.0;
            double s2 = 0.0;
            for (int i = m-1; i >= l; i--) {
               c3 = c2;
               c2 = c;
               s2 = s;
               g = c * e[i];
               h = c * p;
               r = hypot(p,e[i]);
               e[i+1] = s * r;
               s = e[i] / r;
               c = p / r;
               p = c * d[i] - s * g;
               d[i+1] = h + s * (c * g + s * d[i]);

               // Accumulate transformation.

               for (int k = 0; k < n; k++) {
                  h = V[k][i+1];
                  V[k][i+1] = s * V[k][i] + c * h;
                  V[k][i] = c * V[k][i] - s * h;
               }
            }
            p = -s * s2 * c3 * el1 * e[l] / dl1;
            e[l] = s * p;
            d[l] = c * p;

            // Check for convergence.

         } while (abs(e[l]) > eps*tst1);
      }
      d[l] = d[l] + f;
      e[l] = 0.0;
   }

   // Sort eigenvalues and corresponding vectors.

   for (int i = 0; i < n-1; i++) {
      int k = i;
      double p = d[i];
      for (int j = i+1; j < n; j++) {
         if (d[j] < p) {
            k = j;
            p = d[j];
         }
      }
      if (k != i) {
         d[k] = d[i];
         d[i] = p;
         for (int j = 0; j < n; j++) {
            p = V[j][i];
            V[j][i] = V[j][k];
            V[j][k] = p;
         }
      }
   }
}

void EigenvalueDecomposition::orthes () {

   //  This is derived from the Algol procedures orthes and ortran,
   //  by Martin and Wilkinson, Handbook for Auto. Comp.,
   //  Vol.ii-Linear Algebra, and the corresponding
   //  Fortran subroutines in EISPACK.

   int low = 0;
   int high = n-1;

   for (int m = low+1; m <= high-1; m++) {

      // Scale column.

      double scale = 0.0;
      for (int i = m; i <= high; i++) {
         scale = scale + abs(H[i][m-1]);
      }
      if (scale != 0.0) {

         // Compute Householder transformation.

         double h = 0.0;
         for (int i = high; i >= m; i--) {
            ort[i] = H[i][m-1]/scale;
            h += ort[i] * ort[i];
         }
         double g = sqrt(h);
         if (ort[m] > 0) {
            g = -g;
         }
         h = h - ort[m] * g;
         ort[m] = ort[m] - g;

         // Apply Householder similarity transformation
         // H = (I-u*u'/h)*H*(I-u*u')/h)

         for (int j = m; j < n; j++) {
            double f = 0.0;
            for (int i = high; i >= m; i--) {
               f += ort[i]*H[i][j];
            }
            f = f/h;
            for (int i = m; i <= high; i++) {
               H[i][j] -= f*ort[i];
            }
        }

        for (int i = 0; i <= high; i++) {
            double f = 0.0;
            for (int j = high; j >= m; j--) {
               f += ort[j]*H[i][j];
            }
            f = f/h;
            for (int j = m; j <= high; j++) {
               H[i][j] -= f*ort[j];
            }
         }
         ort[m] = scale*ort[m];
         H[m][m-1] = scale*g;
      }
   }

   // Accumulate transformations (Algol's ortran).

   for (int i = 0; i < n; i++) {
      for (int j = 0; j < n; j++) {
         V[i][j] = (i == j ? 1.0 : 0.0);
      }
   }

   for (int m = high-1; m >= low+1; m--) {
      if (H[m][m-1] != 0.0) {
         for (int i = m+1; i <= high; i++) {
            ort[i] = H[i][m-1];
         }
         for (int j = m; j <= high; j++) {
            double g = 0.0;
            for (int i = m; i <= high; i++) {
               g += ort[i] * V[i][j];
            }
            // Double division avoids possible underflow
            g = (g / ort[m]) / H[m][m-1];
            for (int i = m; i <= high; i++) {
               V[i][j] += g * ort[i];
            }
         }
      }
   }
}

void EigenvalueDecomposition::cdiv(double xr, double xi, double yr, double yi) {
   double r,d;
   if (abs(yr) > abs(yi)) {
      r = yi/yr;
      d = yr + r*yi;
      cdivr = (xr + r*xi)/d;
      cdivi = (xi - r*xr)/d;
   } else {
      r = yr/yi;
      d = yi + r*yr;
      cdivr = (r*xr + xi)/d;
      cdivi = (r*xi - xr)/d;
   }
}

void EigenvalueDecomposition::hqr2 () {

   //  This is derived from the Algol procedure hqr2,
   //  by Martin and Wilkinson, Handbook for Auto. Comp.,
   //  Vol.ii-Linear Algebra, and the corresponding
   //  Fortran subroutine in EISPACK.

   // Initialize

   int nn = this->n;
   int n = nn-1;
   int low = 0;
   int high = nn-1;
   double eps = pow(2.0,-52.0);
   double exshift = 0.0;
   double p=0,q=0,r=0,s=0,z=0,t,w,x,y;

   // Store roots isolated by balanc and compute matrix norm

   double norm = 0.0;
   for (int i = 0; i < nn; i++) {
      if (i < low | i > high) {
         d[i] = H[i][i];
         e[i] = 0.0;
      }
      for (int j = fmax(i-1,0); j < nn; j++) {
         norm = norm + abs(H[i][j]);
      }
   }

   // Outer loop over eigenvalue index

   int iter = 0;
   while (n >= low) {

      // Look for single small sub-diagonal element

      int l = n;
      while (l > low) {
         s = abs(H[l-1][l-1]) + abs(H[l][l]);
         if (s == 0.0) {
            s = norm;
         }
         if (abs(H[l][l-1]) < eps * s) {
            break;
         }
         l--;
      }

      // Check for convergence
      // One root found

      if (l == n) {
         H[n][n] = H[n][n] + exshift;
         d[n] = H[n][n];
         e[n] = 0.0;
         n--;
         iter = 0;

      // Two roots found

      } else if (l == n-1) {
         w = H[n][n-1] * H[n-1][n];
         p = (H[n-1][n-1] - H[n][n]) / 2.0;
         q = p * p + w;
         z = sqrt(abs(q));
         H[n][n] = H[n][n] + exshift;
         H[n-1][n-1] = H[n-1][n-1] + exshift;
         x = H[n][n];

         // Real pair

         if (q >= 0) {
            if (p >= 0) {
               z = p + z;
            } else {
               z = p - z;
            }
            d[n-1] = x + z;
            d[n] = d[n-1];
            if (z != 0.0) {
               d[n] = x - w / z;
            }
            e[n-1] = 0.0;
            e[n] = 0.0;
            x = H[n][n-1];
            s = abs(x) + abs(z);
            p = x / s;
            q = z / s;
            r = sqrt(p * p+q * q);
            p = p / r;
            q = q / r;

            // Row modification

            for (int j = n-1; j < nn; j++) {
               z = H[n-1][j];
               H[n-1][j] = q * z + p * H[n][j];
               H[n][j] = q * H[n][j] - p * z;
            }

            // Column modification

            for (int i = 0; i <= n; i++) {
               z = H[i][n-1];
               H[i][n-1] = q * z + p * H[i][n];
               H[i][n] = q * H[i][n] - p * z;
            }

            // Accumulate transformations

            for (int i = low; i <= high; i++) {
               z = V[i][n-1];
               V[i][n-1] = q * z + p * V[i][n];
               V[i][n] = q * V[i][n] - p * z;
            }

         // Complex pair

         } else {
            d[n-1] = x + p;
            d[n] = x + p;
            e[n-1] = z;
            e[n] = -z;
         }
         n = n - 2;
         iter = 0;

      // No convergence yet

      } else {

         // Form shift

         x = H[n][n];
         y = 0.0;
         w = 0.0;
         if (l < n) {
            y = H[n-1][n-1];
            w = H[n][n-1] * H[n-1][n];
         }

         // Wilkinson's original ad hoc shift

         if (iter == 10) {
            exshift += x;
            for (int i = low; i <= n; i++) {
               H[i][i] -= x;
            }
            s = abs(H[n][n-1]) + abs(H[n-1][n-2]);
            x = y = 0.75 * s;
            w = -0.4375 * s * s;
         }

         // MATLAB's new ad hoc shift

         if (iter == 30) {
             s = (y - x) / 2.0;
             s = s * s + w;
             if (s > 0) {
                 s = sqrt(s);
                 if (y < x) {
                    s = -s;
                 }
                 s = x - w / ((y - x) / 2.0 + s);
                 for (int i = low; i <= n; i++) {
                    H[i][i] -= s;
                 }
                 exshift += s;
                 x = y = w = 0.964;
             }
         }

         iter = iter + 1;   // (Could check iteration count here.)

         // Look for two consecutive small sub-diagonal elements

         int m = n-2;
         while (m >= l) {
            z = H[m][m];
            r = x - z;
            s = y - z;
            p = (r * s - w) / H[m+1][m] + H[m][m+1];
            q = H[m+1][m+1] - z - r - s;
            r = H[m+2][m+1];
            s = abs(p) + abs(q) + abs(r);
            p = p / s;
            q = q / s;
            r = r / s;
            if (m == l) {
               break;
            }
            if (abs(H[m][m-1]) * (abs(q) + abs(r)) <
               eps * (abs(p) * (abs(H[m-1][m-1]) + abs(z) +
               abs(H[m+1][m+1])))) {
                  break;
            }
            m--;
         }

         for (int i = m+2; i <= n; i++) {
            H[i][i-2] = 0.0;
            if (i > m+2) {
               H[i][i-3] = 0.0;
            }
         }

         // Double QR step involving rows l:n and columns m:n

         for (int k = m; k <= n-1; k++) {
            bool notlast = (k != n-1);
            if (k != m) {
               p = H[k][k-1];
               q = H[k+1][k-1];
               r = (notlast ? H[k+2][k-1] : 0.0);
               x = abs(p) + abs(q) + abs(r);
               if (x != 0.0) {
                  p = p / x;
                  q = q / x;
                  r = r / x;
               }
            }
            if (x == 0.0) {
               break;
            }
            s = sqrt(p * p + q * q + r * r);
            if (p < 0) {
               s = -s;
            }
            if (s != 0) {
               if (k != m) {
                  H[k][k-1] = -s * x;
               } else if (l != m) {
                  H[k][k-1] = -H[k][k-1];
               }
               p = p + s;
               x = p / s;
               y = q / s;
               z = r / s;
               q = q / p;
               r = r / p;

               // Row modification

               for (int j = k; j < nn; j++) {
                  p = H[k][j] + q * H[k+1][j];
                  if (notlast) {
                     p = p + r * H[k+2][j];
                     H[k+2][j] = H[k+2][j] - p * z;
                  }
                  H[k][j] = H[k][j] - p * x;
                  H[k+1][j] = H[k+1][j] - p * y;
               }

               // Column modification

               for (int i = 0; i <= fmin(n,k+3); i++) {
                  p = x * H[i][k] + y * H[i][k+1];
                  if (notlast) {
                     p = p + z * H[i][k+2];
                     H[i][k+2] = H[i][k+2] - p * r;
                  }
                  H[i][k] = H[i][k] - p;
                  H[i][k+1] = H[i][k+1] - p * q;
               }

               // Accumulate transformations

               for (int i = low; i <= high; i++) {
                  p = x * V[i][k] + y * V[i][k+1];
                  if (notlast) {
                     p = p + z * V[i][k+2];
                     V[i][k+2] = V[i][k+2] - p * r;
                  }
                  V[i][k] = V[i][k] - p;
                  V[i][k+1] = V[i][k+1] - p * q;
               }
            }  // (s != 0)
         }  // k loop
      }  // check convergence
   }  // while (n >= low)

   // Backsubstitute to find vectors of upper triangular form

   if (norm == 0.0) {
      return;
   }

   for (n = nn-1; n >= 0; n--) {
      p = d[n];
      q = e[n];

      // Real vector

      if (q == 0) {
         int l = n;
         H[n][n] = 1.0;
         for (int i = n-1; i >= 0; i--) {
            w = H[i][i] - p;
            r = 0.0;
            for (int j = l; j <= n; j++) {
               r = r + H[i][j] * H[j][n];
            }
            if (e[i] < 0.0) {
               z = w;
               s = r;
            } else {
               l = i;
               if (e[i] == 0.0) {
                  if (w != 0.0) {
                     H[i][n] = -r / w;
                  } else {
                     H[i][n] = -r / (eps * norm);
                  }

               // Solve real equations

               } else {
                  x = H[i][i+1];
                  y = H[i+1][i];
                  q = (d[i] - p) * (d[i] - p) + e[i] * e[i];
                  t = (x * s - z * r) / q;
                  H[i][n] = t;
                  if (abs(x) > abs(z)) {
                     H[i+1][n] = (-r - w * t) / x;
                  } else {
                     H[i+1][n] = (-s - y * t) / z;
                  }
               }

               // Overflow control

               t = abs(H[i][n]);
               if ((eps * t) * t > 1) {
                  for (int j = i; j <= n; j++) {
                     H[j][n] = H[j][n] / t;
                  }
               }
            }
         }

      // Complex vector

      } else if (q < 0) {
         int l = n-1;

         // Last vector component imaginary so matrix is triangular

         if (abs(H[n][n-1]) > abs(H[n-1][n])) {
            H[n-1][n-1] = q / H[n][n-1];
            H[n-1][n] = -(H[n][n] - p) / H[n][n-1];
         } else {
            cdiv(0.0,-H[n-1][n],H[n-1][n-1]-p,q);
            H[n-1][n-1] = cdivr;
            H[n-1][n] = cdivi;
         }
         H[n][n-1] = 0.0;
         H[n][n] = 1.0;
         for (int i = n-2; i >= 0; i--) {
            double ra,sa,vr,vi;
            ra = 0.0;
            sa = 0.0;
            for (int j = l; j <= n; j++) {
               ra = ra + H[i][j] * H[j][n-1];
               sa = sa + H[i][j] * H[j][n];
            }
            w = H[i][i] - p;

            if (e[i] < 0.0) {
               z = w;
               r = ra;
               s = sa;
            } else {
               l = i;
               if (e[i] == 0) {
                  cdiv(-ra,-sa,w,q);
                  H[i][n-1] = cdivr;
                  H[i][n] = cdivi;
               } else {

                  // Solve complex equations

                  x = H[i][i+1];
                  y = H[i+1][i];
                  vr = (d[i] - p) * (d[i] - p) + e[i] * e[i] - q * q;
                  vi = (d[i] - p) * 2.0 * q;
                  if (vr == 0.0 & vi == 0.0) {
                     vr = eps * norm * (abs(w) + abs(q) +
                     abs(x) + abs(y) + abs(z));
                  }
                  cdiv(x*r-z*ra+q*sa,x*s-z*sa-q*ra,vr,vi);
                  H[i][n-1] = cdivr;
                  H[i][n] = cdivi;
                  if (abs(x) > (abs(z) + abs(q))) {
                     H[i+1][n-1] = (-ra - w * H[i][n-1] + q * H[i][n]) / x;
                     H[i+1][n] = (-sa - w * H[i][n] - q * H[i][n-1]) / x;
                  } else {
                     cdiv(-r-y*H[i][n-1],-s-y*H[i][n],z,q);
                     H[i+1][n-1] = cdivr;
                     H[i+1][n] = cdivi;
                  }
               }

               // Overflow control

               t = fmax(abs(H[i][n-1]),abs(H[i][n]));
               if ((eps * t) * t > 1) {
                  for (int j = i; j <= n; j++) {
                     H[j][n-1] = H[j][n-1] / t;
                     H[j][n] = H[j][n] / t;
                  }
               }
            }
         }
      }
   }

   // Vectors of isolated roots

   for (int i = 0; i < nn; i++) {
      if (i < low | i > high) {
         for (int j = i; j < nn; j++) {
            V[i][j] = H[i][j];
         }
      }
   }

   // Back transformation to get eigenvectors of original matrix

   for (int j = nn-1; j >= low; j--) {
      for (int i = low; i <= high; i++) {
         z = 0.0;
         for (int k = low; k <= fmin(j,high); k++) {
            z = z + V[i][k] * H[k][j];
         }
         V[i][j] = z;
      }
   }
}

 EigenvalueDecomposition::EigenvalueDecomposition (Matrix Arg) {
   double** A = Arg.getArray();
   n = Arg.getColumnDimension();
   V = new double*[n];
   for (int i = 0; i < n; i++) {
       V[i] = new double[n];
   }
   d = new double[n];
   e = new double[n];

   issymmetric = true;
   for (int j = 0; (j < n) & issymmetric; j++) {
      for (int i = 0; (i < n) & issymmetric; i++) {
         issymmetric = (A[i][j] == A[j][i]);
      }
   }

   if (issymmetric) {
      for (int i = 0; i < n; i++) {
         for (int j = 0; j < n; j++) {
            V[i][j] = A[i][j];
         }
      }

      // Tridiagonalize.
      tred2();

      // Diagonalize.
      tql2();

   } else {
      H = new double*[n];
      for (int i = 0; i < n; i++) {
          H[i] = new double[n];
      }
      ort = new double[n];

      for (int j = 0; j < n; j++) {
         for (int i = 0; i < n; i++) {
            H[i][j] = A[i][j];
         }
      }

      // Reduce to Hessenberg form.
      orthes();

      // Reduce Hessenberg to real Schur form.
      hqr2();
   }
}

EigenvalueDecomposition::~EigenvalueDecomposition() {
    delete[] V;
    delete[] d;
    delete[] e;
    delete[] H;
    delete[] ort;
}

Matrix EigenvalueDecomposition::getV () {
   return Matrix(V,n,n);
}

double* EigenvalueDecomposition::getRealEigenvalues () const {
   return d;
}

double* EigenvalueDecomposition::getImagEigenvalues () const {
   return e;
}

Matrix EigenvalueDecomposition::getD () {
   Matrix X = Matrix(n,n);
   double** D = X.getArray();
   for (int i = 0; i < n; i++) {
      for (int j = 0; j < n; j++) {
         D[i][j] = 0.0;
      }
      D[i][i] = d[i];
      if (e[i] > 0) {
         D[i][i+1] = e[i];
      } else if (e[i] < 0) {
         D[i][i-1] = e[i];
      }
   }
   return X;
}
